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Erdős Problem #848

Erdős problem #848 · erdosproblems.com/848

Is the maximum size of a set A⊆{1,…,N}A\subseteq \{1,\ldots,N\} such that ab+1ab+1 is never squarefree (for all a,b∈Aa,b\in A) achieved by taking those n≡7(mod25)n\equiv 7\pmod{25}? Resolved for all sufficiently large NN: any near-maximal AA is contained in {n≡7(mod25)}\{n\equiv 7\pmod{25}\} or {n≡18(mod25)}\{n\equiv 18\pmod{25}\}, leaving only a finite check.

Result
Proved(see note)
Status
Resolved
AI contribution
AI co-developed
Method
Argument
Field
Number Theory
Posed by
Paul Erdős, András Sárközy
Year posed
1992
Years open
33y
Solved
2025-11-20
Model
GPT-5
Vendor
OpenAI
Collaborators
Mehtaab Sawhney, Mark Sellke
Verification
Site-confirmed
Publication
Announced
Significance
10 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Resolved for all sufficiently large N via a stability theorem; small N remain a finite computation (erdosproblems.com marks the problem DECIDABLE)

What the AI did

Sawhney's note resolving the problem cites a ChatGPT (GPT-5) session in the provenance of the key lemma, and Tao's AI-contributions ledger records the solve as GPT-5 working with Sawhney and Sellke (October-November 2025).

Verification

erdosproblems.com marks the problem resolved up to a finite check and links Sawhney's note; no formal artifact and no journal review.

Sources

Discussion